Let X1, X2 be independent exponential random variables with common mean . Show that the Legendre transformation

Let X1, X2 be independent exponential random variables with common mean μ. Show that the Legendre transformation of the joint cumulant generating function is K∗ (x1, x2; μ) =

x1+x2−2μ

μ − log (

x1

μ ) − log (

x2

μ ).

Show also that the Legendre transformation of the cumulant generating transformation of X̄ is K∗ (x; μ) = 2 (

x−μ

μ ) − 2 log (

x

μ ).

Hence derive the double saddlepoint approximation for the conditional distribution of X1 given that X1 + X2 = 1. Show that the re-normalized double saddlepoint approximation is exact.